mellinger lesson 7 lvg model & x co toshihiro handa dept. of phys. & astron., kagoshima...
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Mellinger
Lesson 7LVG model & XCO
Toshihiro HandaDept. of Phys. & Astron., Kagoshima University
Kagoshima Univ./ Ehime Univ.Galactic radio astronomy
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Multi-line observations(1)
▶ LTE approximation■ Tex is constant between any two levels
■ Line intensities differ due to TB=Tex (1-e-)
■ Compare lines with ≫1 and≪1
TB,thick=Tex, TB,thin=Tex,
▶ Optical depth from intensity→column density
▶ Optically thick line→excitation temperature
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Multi-line observations(2)
▶ Multi-levels (allow j=±1: diatomic mol.)dnj=nj+1Aj+1,j-njBj, j+1Ij+1,j+nj+1Bj+1,jIj+1,j-njCj,j+1 +nj+1Cj+1,j
n=nj total number is const.
■ Solve it under steady state dnj=0
▶ Change of Ij+1,j:simliar to the 2 level model
= (h)/(4) () nj Aj+1,j
= (h)/(4) () (nj Bj,j+1-nj+1 Bj+1,j)
▶ Change of intensity dI=(–I)dx■ Depend on the large scale structure of the cloud
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LVG model(1)
▶ Assume: large, monotonic vel. grad.■ Radiative coupling is bounded in local.
▶ Assume: abs. and rad. are thermally coupled.■ escape probability ▶ Ii,j = (1-i,j ) Si,j+i,j B(TCMB)
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LVG model(2)
▶ Optical depth →photon escape prob. controlled by geometrical structure“Abs. & rad. are bounded in a small space”
→Vel. structure has large gradient. (LVG)
■ =[1-exp(-)]/: 1D model = slab■ =[1-exp(-)]/(3): spherical symmetric
▶ LVG model■ Under this structure, derive the all level population■ Tex for each trans. are fixed.→intensity of each line
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LVG model (3)
▶ Three input parameters■ Kinetic temp. of H2 Tk
■ Gas density of H2 n(H2)
■ Mol. Numb per depth & velocity span n(X)/(dv/dr)
▶ Solve equations numerically■ Goldreich & Kwan (1974) ApJ 189, 441■ Scoville & Solomon (1974) ApJ 187, L67
Scoville&Solomon (1974)
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Features of a molecular cloud
▶ Features of an actually observed emission line
▶ Gaussian like profile
▶ width: much wider than thermal motion■ Larger scale motion than thermal■ Turbulence?
▶ Intensity: much colder than gas temperature■ beam filling factor
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Turbulent model
▶ Motion in a beam (observed pixel)■ Gaussian like velocity field: random motion■ wider than thermal width→supersonic turbulence
Problem: rapid dissipationWhat supplies the turbulent motion energy?
■ Super high reso. obs: thermal width is observed!
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Feasibility of the LVG model
▶ The line width is finite!■ Disconnect if velocity difference is large.■ Only a small region is connected by radiation.
▶ Order is OK with LVG, even diff. geometry.■ We cannot know the detail geom. structure!
▶ If you want to calculate more precisely, ■ e.g. photon tracing using Monte-Carlo simulation
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Beam filling factor(1)
▶ Observed resolution is poor.
▶ Inhomogeneous gas in a beam■ First approx. : all or nothing
obs. Beam sizegas is located only here.
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Beam filling factor(2)
▶ Beam filling factor■ The more parameters, the more freedom.■ Filling factor may be different for diff. lines.
Oh! More freedom!!We need the simplest model
■ The same factors give no effect on line ratio!The effective critical densities are close.
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Geometrical structure of a cloud
▶ No information assume spherical symmetry■ “Common sense” in astronomy, 1D approx.
▶ Actually far from a spherical geometry■ “infinite” fine structure. fractal structure■ filaments
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Volume filling factor
▶ In the “outer boundary” of a cloud
▶ Inhomogeneous gas in a cloud■ First approx. : all or nothing■ clumpy model
gas is located only here =clumpVolume of a cloud
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Conversion factor X (1)
▶ CO intensity ∝ gas column density■ Why? CO is optically thick!!
Intensity ratio is far from abundance13CO/12CO intensity ratio ~ 10-1
13CO/12CO abundance ~ 1/89( 太陽系 ) 、 1/67(MWG)
■ Empirical relation originallyLine profiles are similar in 12CO and 13CO.
▶ N(H2)=X ∫ TB(CO,J=1-0) dv■ X=2.3×1020 cm-2/(K km s-1)
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Conversion factor X (2)
■ Gamma ray: interaction between CR & proton■ Correlation between gamma, HI, and CO
CGRO, NASADicky&Lockman HI
AMANOGAWA CO
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Conversion factor X (3)
▶ Why it works well?■ Cloud property is similar over the galaxy(?)■ TB gives beam filling factor(?)
←small beam filling factor in general
▶ “theoretical” model■ Virial equiv.→ m∝ Rv2, optically thick→TB∝ Tex
■ In this case, X=N(H2)/(∫TB dv)∝ n(H2)1/2r3/2/Tex
■ subthermally excited→Tex∝ n(H2)-1/2: LVG model■ ∴ If clump size is const., X is const.